Detection and Characterization of Chemical Vapor. Fugitive Emissions by Nonlinear Optimal Estimation: Theory and Simulation

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1 Detection and Characterization of Chemical Vapor Fugitive Emissions by Nonlinear Optimal Estimation: Theory and Simulation Christopher M. Gittins Physical Sciences Inc., 0 New England Business Center, Andover, MA 01810, USA Gittins@psicorp.com This paper addresses detection and characterization of chemical vapor fugitive emissions in a non-scattering atmosphere by processing of remotely-sensed long wavelength infrared spectra. The analysis approach integrates a parameterized signal model based on the radiative transfer equation with a statistical model for the infrared background. The maximum likelihood model parameter values are defined as those which maximize a Bayesian posterior probability and are estimated using a Gauss-Newton algorithm. For algorithm performance evaluation we simulate observation of fugitive emissions by augmenting plume-free measured spectra with synthetic plume signatures. As plumes become optically-thick, the Gauss-Newton algorithm yields significantly more accurate estimates of chemical vapor column density and significantly more favorable plume detection statistics than clutter-matched-filter-based and adaptive-subspace-detector-based plume characterization and detection. OCIS codes: Radiative transfer 1

2 Multispectral and hyperspectral imaging Machine vision algorithms Air pollution monitoring Passive remote sensing Spectroscopy, infrared. 1. Introduction Long wavelength infrared (LWIR) spectrometry has been employed for several decades for remote sensing of chemical vapor fugitive emissions. This paper addresses detection and characterization of fugitive emissions in a non-scattering atmosphere by processing of LWIR spectra collected in an open path configuration. The analysis approach employs a parameterized signal model derived from the radiative transfer equation (RTE) and applies estimation theory to determine the maximum likelihood parameter values. In this regard, it bears some similarity to estimation methods developed for atmospheric profile retrieval using data collected by atmospheric sounding spectrometers. In both cases, the at-sensor spectral radiance is a nonlinear function of atmospheric temperature and constituent profiles and retrieval of the properties of interest requires inverse solution of the RTE. Because inverse solution of the RTE is mathematically ill-posed, the physical properties of interest must be inferred using estimation theory. The use of estimation theory for atmospheric profile retrieval was demonstrated by Rodgers in 1976 [1]. Variations on and refinements of Rodgers implementation have been made by numerous authors in the decades since, see, e.g., [-7]. The estimation algorithm presented here has characteristics in common with methods described in those publications. The RTE used here is well-suited for fugitive emissions detection from short range (<5 km) where

3 observations are made with a horizontal or near-horizontal line-of-sight to the area of interest. That noted, the nonlinear estimation approach is generally-applicable and may be adapted to address other observation scenarios by modifying the RTE and/or the statistical model for the background. We define the maximum likelihood model parameter values as those which maximize a Bayesian posterior probability. The posterior probability distribution function (pdf) is the product of a conditional pdf for the observed spectrum and a prior pdf for infrared background spectra. Accounting for infrared background characteristics using a prior pdf rather than a physics-based model is both statistically-justifiable, i.e., no information is lost in an information theoretical sense, and a practical necessity in order to make the estimation problem computationally tractable. The signal model for measured spectra is a nonlinear function of the model parameters and we use a Gauss-Newton algorithm to determine the maximum likelihood parameter values. In addition to presenting the signal model and parameter estimation algorithm, we address algorithm initialization and derive the uncertainties associated with the estimated parameter values. For algorithm validation, we simulate observation of fugitive emissions by augmenting plume-free measured spectra with synthetic plume signatures using a process which preserves the noise characteristics of the original data. We compare the results obtained by processing the simulated data with the Gauss-Newton algorithm with those obtained by processing the data with an adaptive subspace detector (for plume detection) and a cluttermatched filter (for estimation of path-integrated vapor concentration), both of which are derived from a linear additive signal model which follows from an approximation to the underlying RTE [8-13]. The purpose of simulation in this work is not to test the validity of the clear air 3

4 approximation but, postulating that the underlying atmospheric model is correct for the cases evaluated, to assess the effect of using the approximate, linearized RTE rather than the exact RTE as the basis for chemical vapor detection and characterization. Comparing results obtained for simulated observations in a uniform, non-scattering atmosphere is the simplest test case. Readers will note that the signal model used for nonlinear estimation is easily modified to address observations in more complex atmospheres. While nonlinear estimation and estimation based on the linear, additive signal model produce essentially identical results when applied to optically-thin plumes, nonlinear estimation provides significantly more accurate estimates of path-integrated chemical vapor concentration (column density) and significantly more favorable detection statistics when vapor plumes are optically-thick at one or more of the sensor observation wavelengths. Simulations indicate that nonlinear estimation utilizing a Gauss-Newton algorithm can reduce column density estimation error by an order of magnitude and reduce false alarm rates by several orders of magnitude relative to the clutter-matched filter for a plume optical density as low as 1.0, i.e., ( ) transmissi on = exp 1.0 at the wavelength of strongest absorption. As expected, the performance of the Gauss-Newton algorithm relative to the linear estimators improves with increasing plume optical density, i.e., as the optically-thin plume approximation breaks down.. Algorithm Formulation.1. Radiative Transfer Model The radiative transfer model which underlies the analysis approach presented here follows from the stratified atmosphere approximation: the atmosphere along the sensor s line-of-sight is modeled as n layers with each layer having uniform temperature, pressure, and chemical 4

5 composition [14]. The concept is illustrated in Figure 1. Under this model, the monochromatic at-sensor spectral radiance at the interface of Layer i and Layer i+1 is [15]: R i ( λ) = R ( λ) τ ( λ) + L ( λ) [ 1 τ ( λ) ] i 1 i i i (1) where τ i is the transmission of layer i and L i is the blackbody (Planck) function evaluated at the thermodynamic temperature of Layer i, T i. If R 0 arises from a solid background then it is the surface leaving radiance, i.e., R 0 includes reflected downwelling radiance as well as surface emission, not simply the blackbody function evaluated at the surface temperature. When a vapor plume is present at Layer p, the at-sensor spectral radiance is: R s ( τ ) L + τ [ τ R + ( 1 ) L ] = 1 τ () a a a where τ a is the atmospheric transmission from the sensor along the line-of-sight to the plume, L a is the path-weighted blackbody radiance from the sensor to Layer p, τ p is the plume transmission, R b is the background spectral radiance incident on the plume layer, and p b p p L p is the blackbody function evaluated at the plume temperature. (Note that the wavelength dependence of the quantities in Eq. () has been suppressed for clarity.) The plume transmission follows Beer s law. In this work we consider a plume consisting of a single vapor and [ ρ κ ] τ p = exp (3) where κ is the absorption cross-section of the chemical vapor and ρ is its column density. Absorption cross-sections are temperature- and pressure-dependent; however, those dependencies are weak near 300 K and 1 atm and we ignore them here. Eq. () has formed the basis for multiple fugitive emissions detection and characterization approaches, see, e.g., [8,16-18]. When the atmosphere is homogeneous along the line-of-sight to the plume, L a is simply the blackbody function evaluated at the air 5

6 temperature. Methods for calculating L a along an inhomogeneous path are described by Clough, et al. [19] and Rodgers [15]. In those treatments, L a corresponds to the blackbody function evaluated at an effective temperature derived from the characteristics of the path along the sensor s line-of-sight. Although treating the atmosphere as a single uniform layer here may appear to be a crude approximation likely to introduce significant error in estimated plume characteristics, Sheen, et al. s analysis of background and atmospheric variability effects [0] suggests that this is generally not the case for short range detection: background variability is generally the primary source of uncertainty and error. Following Eqs. (1) and (), the at-sensor radiance in the absence of the plume is: R ( 0) s = ( 1 τ a ) La + τ a Rb (4) and the change in at-sensor radiance due to the plume is: R s R ( 0) ( 0) = ( τ ) [( L R ) + τ ( L L )] s 1 (5) p a Note that when the temperature of the plume is equal to the atmospheric temperature then L L = 0 and the change in at-sensor radiance is independent of the atmospheric transmission, p a i.e., atmospheric compensation is not necessary to estimate plume properties. For the remainder of this document we consider the case where the air temperature is uniform and known and the plume temperature is equal to the air temperature. (In general, atmospheric transmission effects may be ignored when the effective thermal contrast between the plume and the air is much less than the effective thermal contrast between the air and the background.) These constraints are not essential for the analysis approach to be valid; however, they facilitate a more concise presentation of the analysis approach. We note in the text where relaxing these constraints affects the details of the parameter estimation algorithm. In practice, if it is necessary or desirable to presume a known and uniform plume temperature, there are s a p a 6

7 methods for making a reasonable estimate. The simplest approach is to presume that it is equal to the local atmospheric temperature. Alternatively, depending upon the characteristics of the measured radiance spectra, the atmospheric temperature in the vicinity of the plume may be estimated from the spectral radiance at wavelengths where atmospheric water vapor (or other atmospheric constituent which is reasonably well-mixed over the sensor s line-of-sight) becomes optically-thick over a range comparable to the estimated distance to the plume... Sensor Signal Model The equations in the preceding section apply to monochromatic radiation. In developing the signal model for describing measured spectra it is necessary to address the effects of the sensor s spectral resolution. In the absence of measurement noise, the apparent spectral radiance at acquisition wavelength λ s is: x ( λ ) R ( λ) g( λ λ ) s = s ; s dλ (6) g λ, λ s, is where R ( λ) is as per Eq. (4) or (5), and the instrument lineshape function, ( ) normalized such that ( λ; λ ) dλ = 1 g s. Combining Eqs. (5) and (6), the change in sensormeasured spectral radiance due to the plume is: x ( λ ) x ( λ ) = 1 τ ( λ) ( 0 [ ] ( ) ) p [ L a λ Rs ( λ) ] g( λ λs ) 0 s dλ (7) p s, where x p and x 0 are the band-averaged spectral radiances as per Eq. (6) with and without the plume present, respectively. In order to facilitate signal model parameterization and computationally-efficient parameter estimation, we approximate Eq. (7) as: where L ( λ ) x p [ ] [ L ] = and the effective plume transmission is a L a s x0 = 1 τ x 0 (8) e a 7

8 ( α ρ κ ) τ e = exp 0 (9) The quantity κ is the vapor absorption cross-section averaged over the instrument lineshape κ ( λ ) κ( λ) g( λ λ ) s =, s dλ. (10) The quantity ρ 0 is a reference column density defined to make α a unitless quantity, α ρ ρ 0. We choose ρ 1 max{ κ( λ) } 0 = so that α corresponds to the plume optical density at the strongest absorption feature in the high resolution spectrum. The approximation in Eq. (9) is addressed further in Section 4. Parameterization of the plume-free background spectral radiance, x 0, is addressed in the following Section. From this point forward, measured spectra are treated as k-dimensional vectors: ~ x = x + e (11) where x denotes the vector of the noise-free spectral radiance from Eq. (6) and e denotes measurement noise. (The tilde denotes a noisy measurement.) In developing the analysis approach, we presume that the measurement noise is normally-distributed with zero mean and is uncorrelated from band to band, e ( 0, D) ~ N where D is a diagonal matrix. The diagonal elements of D are the 1σ standard deviation in each band resulting from sensor noise and spectral clutter. We do not presume that the noise variance is equal in all bands..3. Infrared Background Model There are many reasonable approaches to modeling and parameterization of infrared backgrounds. We use a factor analysis-based model [1] because it facilitates definition of a Gaussian prior pdf for model parameter values. This in turn simplifies implementation of the parameter estimation algorithm. The model is summarized in Appendix A and is similar to the Probabilistic Principal Components model described by Tipping and Bishop []. Briefly, the 8

9 model presumes that, in the absence of a vapor plume, spectra may be described by a linear mixing model: x = µ + Bβ (1) where µ is the mean background spectrum, B is the k m dimensional matrix whose columns are the basis vectors used to span the data space and β is an m 1 vector of weight coefficients. The key detail on the implementation of Eq. (1) is that the β vectors for the sample are presumed to be uncorrelated, have zero mean and unit standard deviation: ( 0, ) β ~ N I m (13) where I m is the m m identity matrix. Eq. (13) is exact for Principal Components Analysis (PCA) applied to multivariate normal distributed data. The B matrix follows from a regularization approximation of the calculated sample covariance matrix and is calculated using eigenvalues and eigenvectors from a PCA of the data. Calculation of the B matrix and estimation of the model order, m, are described in Appendix A..4. Model Parameter Estimation Following Eqs. (8), (9), and (1), the signal model parameters are the chemical vapor optical density, the weight coefficients for the basis vectors for the background (i.e., the elements of the β vector), and the plume/atmospheric temperature. For each spectrum, the parameters define a vector θ, θ = [ α,β,t a ]. We take a Bayesian approach to estimating the maximum likelihood parameter values. Given an observation, x ~, the probability that the parameter values are θ is ( ~ x θ) p( θ) ( θ ~ p p x) = p ~ (14) ( x) 9

10 where ( x~ θ ) p is the conditional probability of observing ~ x given θ, ( θ) p is the prior probability of the parameter values being θ, and p ( x~ ) is the prior probability of observing ~ x. The maximum likelihood parameter values are those which maximize p ( x~ θ ) or, equivalently, those which minimize ln p( x~ θ ). The maximum likelihood model parameters are denoted θˆ. Following Eq. (8), the model function for x is: f ( θ) τ e o x + [ 1 τ e ] o L a = 0 (15) where the vector x 0 is the estimated noise-free background spectrum given by Eq. (1), 1 is a k-element vector of ones, τ e is the plume transmission calculated using Eq. (10), L a is the blackbody function evaluated at the plume temperature, and denotes the element-by-element (Hadamard) product of the two vectors. In order to facilitate computationally-efficient parameter estimation, we postulate that p ( x~ θ ) follows a multivariate normal distribution: where ln p 1 ( ~ x θ) = ~ x f ( θ) 1 D is a noise-whitening matrix and x θ T 1 [ ] D [ ~ x f ( θ) ] + cx θ c is a constant; = diag{ σ σ σ } (16) D 1,,..., k, whereσ i is the standard error of the measured spectral radiance in band i. To deal with the p ( θ) term in Eq.(14) we postulate that θ can partitioned into a subset of parameters where the prior follows a multivariate normal distribution, i.e., the parameter values are constrained, and a subset where the prior is uniform, i.e., the parameter values are unconstrained. Following this presumption 1 T ln p( θ) = [ θc θ a ] Sθ [ θ c θ a ] + cθ (17) 10

11 where θ c is the subset of θ which are constrained and c θ is a constant. Specification of θ c is described below. The vector θ a is the a priori estimate of θ c and S θ is a regularization matrix which penalizes deviations about θ a. For the purpose of estimating the maximum likelihood model parameter values, it is not necessary to know the details of p ( x~ ) because it is simply a normalization factor and is independent of θ. Following Eqs. (14), (16), and (17), the model parameter values which maximize p ( θ ~ x) are also those which minimize the cost function: C = [ ~ T 1 T x f ( θ) ] D [ ~ x f ( θ) ] + [ θ θ ] S [ θ θ ] c a θ c a (18) The first term on the righthand side of Eq. (18) penalizes deviations between the measured spectrum and the model spectrum. The second term on the righthand side of Eq. (18) penalizes deviations of the constrained model parameters from their nominal values. As stated above, in order to demonstrate the nonlinear estimation approach without making the mathematics unnecessarily complicated, we consider parameter estimation when the plume temperature is known and is equal to the effective temperature of the atmosphere. Also, we presume that if a chemical vapor cloud is present in the scene that its location and optical density are a priori unknown and that optical density values are independent of background characteristics. Following these presumptions, α is treated as an unconstrained parameter, θ = c β, and second term on the righthand side of Eq. (18) reduces to T T [ θ θ ] S [ θ θ ] β β θ (19) c a c a = With respect to Eqs.(17)-(19), while α cannot be <0 or exceed the value corresponding to the atmospheric pressure and, in principle, should be constrained, in practice it is more 11

12 computationally-efficient to leave α unconstrained and then reject physically implausible values after the parameter estimation algorithm terminates. The maximum likelihood model parameter values following from Eq.(18) may be estimated using a Gauss-Newton algorithm after expressing the cost function in quadratic form C = r T r (0) where r is a p-dimensional vector. The Gauss-Newton algorithm updates parameter values iteratively as θ i+ 1 i T 1 T ( J i J i ) J i ri = θ (1) where i indicates iteration number and J is the Jacobian of the r vector, r θ. Eq. (1) is a general result. The dimensionality of the r vector and the Jacobian depends upon the details of the signal model and the cost function. Following Eqs.(18) and (19), the r vector in this case consists of (k+m) elements, 1/ [ D [ ~ x f ( θ) ] β] r = ; () The Jacobian of r is a (k + m) (m + 1) matrix which is the concatenation of its partial derivative with respect to the β parameters and its partial derivative with respect to the α parameter: r r J = ; (3) β α where r β [ r β r β ;...; r ] 1 ; β m is a (k + m) m matrix: r β D = 1/ diag I m { τ } e B (4) 1

13 and r α is a (k+m) 1 column vector. The quantity diag{ τ e } is a k k matrix with τ e on the diagonal and zeros on the off-diagonal; { τ } B identity matrix. The r α vector is: diag e is a k m matrix and m I is the m m D = α diag 0 { τ o s} δ 1/ r e 0 m (5) where s ρ κ, δ 0 = L a ( µ + Bβ) is the radiance contrast between the air and the estimated = 0 background spectrum (a k 1 column vector) and 0 m is an m 1 vector of zeros. The formulation of Eqs. (3)-(5) ensures that J is invertible under virtually all T i J i physically plausible detection scenarios and thereby makes Eq. (1) extremely stable. (The I m matrix in Eq. (4) is the principal source of stability.) When the atmospheric temperature is treated as a fixed parameter, the only scenario where J is guaranteed not to be invertible, and T i J i therefore Eq. (1) cannot produce an accurate estimate of plume column density, is when the plume is opaque at all wavelengths, i.e., τ = 0. The uncertainty associated with the estimated α e value tends to infinity as δ 0 0 ; however, this does not prevent the algorithm from converging. Although the Eqs.()-(5) follow from the presumption that the atmospheric temperature is known, the framework above permits it to be treated as an estimated parameter. For example, it may be treated as a constrained parameter by augmenting the r vector with a T T σ term ( a 0 ) T, where T 0 is an a priori estimated air temperature and σ T is the uncertainty in the estimated temperature. In principle, T a may also be treated as an unconstrained parameter; however, in the absence of a constraint, r T 0 as α 0. This results in J T J a being ill- 13

14 conditioned as α 0 (and non-invertible for α = 0 ) thereby rendering application of the Gauss-Newton algorithm problematic (or impossible). Treating T a as a free parameter is only effective when the vapor cloud is optically-thick, in which case the measured spectral radiance at the wavelength(s) of peak optical density provides a reasonable measure of the plume temperature..5. Algorithm Initialization The Gauss-Newton algorithm requires an initial guess at the maximum likelihood model parameters. We make our initial guess by applying several approximations to Eq. (15) in order to create a linear additive signal model. The linear additive model facilitates direct calculation of the maximum likelihood model parameters by matrix algebra. The first step in developing the approximate model is to presume an optically-thin vapor plume, α << 1, and approximate Eq. (15) as x p ( L ) = x0 + s o a x 0 α (6) By further presuming that the plume is viewed against a blackbody background, Eq. (6) simplifies to: where the vector s ' is and x p = x 0 + α' s' (7) dl a s' = s o T 0 dt (8) T a T eff α ' = α (9) T 0 14

15 ] Ta The quantity [ d L / dt is the derivative of the blackbody function with respect to temperature evaluated at the air temperature, a Teff is effective thermal contrast between the air temperature and the radiometric temperature of the background, and T0 is a reference thermal contrast, nominally 1 K. Combining Eqs. (1) and (7) we obtain the approximate signal model: By replacing f ( θ) in Eq. (15) with ( θ' ) ( θ ) = ' s' + Bβ' µ g ' α + (30) g above and maintaining the constraint in Eq. (1), then applying the criterion that θ C 0 at the minimum of the cost function, we obtain a linear ' = system of equations which may be solved directly for the maximum likelihood values of α ' and β ', ˆα ' and ˆβ ' : T ˆ' α s' D = ˆ' B 1 s' s' T D 1 B 1 ~ ( x µ ) ( ~ x µ ) 1 T 1 T 1 β D s' Λ m D T D s' B (31) The matrix to be inverted consists of four sub-blocks: s' T D 1 s' is a 1 1 sub-block, s' T D 1 B is T 1 T 1 a 1 m sub-block, ' ( ' ) T B D s = s D B is an m 1 sub-block, and the Λ m sub-block is an m m diagonal matrix whose non-zero elements are the leading m eigenvalues of the noisewhitened sample covariance matrix, 1/ D ΣD 1/. The system of equations in Eq. (31) yields the clutter-matched filter result [9,1,13]: T ˆ 1 s' Σ ( ~ x µ ) ˆ' α = (3) T ' ˆ 1 s Σ s' when Σ ˆ T = BB + D is the regularized sample covariance matrix. Using the relation α = 0 ( ) α' T T eff, enables comparison of column densities estimated using the linearized 15

16 model in Eq. (30) with the results obtained presuming the RTE-based signal model in Eq. (15). Calculation of Teff is addressed below..6. Uncertainty Analysis It is instructive to compare the Cramer-Rao lower bound (CRLB) on the uncertainty in the column density estimated using the Gauss-Newton solver with that associated with the linear model estimate. The CRLB on the uncertainties in parameters determined using the Gauss-Newton algorithm may be determined from the elements of the inverse of the Fisher information matrix [3]: [ ( ˆ )] 1 θ i [ I ( θˆ )] ii σ (33) where σ ( a) is the 1σ standard deviation the quantity a. Following Eqs. (14), (18) and (19), the elements of the Fisher information matrix are: [ I() θˆ ] ij ( θ ~ x) ln p = E θ i θ j T [ J J] ij (34) Combining Eqs. (33) and (34), T 1 [ ] ii [ ( ˆ α )] = ( J J) σ (35) The index i in Eq. (35) corresponds the index of α in the parameter vector. The matrix ( ) 1 J T J is calculated with each iteration of the Gauss-Newton algorithm so σ ( ˆ α ) may be determined for each parameter estimated with no additional computational expense. For the linear model given by Eq. (30), there exists a closed-form expression for [ ( ˆα )] σ : 16

17 α α ' [ σ ( ˆ α )] = [ σ ( ˆ' α )] L T0 = T α + ( ) T [ σ ( ˆ' α )] + ( ˆ' α ) [ σ ( T )] ( T ) σ T (36) where [ α α' ] and [ α ( T )] are from Eq. (9). The subscript L in Eq. (36) is to distinguish the uncertainty derived from the linear model from that estimated using the Gauss-Newton solver. The quantity σ ( ˆα' ) is the CRLB associated with the Gaussian probability distribution function which follows from the cost function defined using the linear model: 1 [ ( ˆ' α )] [ s' D s] T 1 σ (37) When there is no uncertainty in the thermal contrast between the air and the background, i.e., when the plume and the air temperature are known precisely, ( T ) = 0 to: 0 1 [ ( ˆ α )] = [ s' D s' ] T 1 σ and Eq. (36) simplifies T σ L (38) Teff Note that the uncertainty in the estimated column density tends to infinity as T 0. The effective thermal contrast in Eq. (39) is that which minimizes the sum-squared deviation between ( L ) s o in Eq. (6) and s' in Eq. (7): a x 0 eff T eff = T 0 T ( s' ) [ s o ( L a x 0 )] T ( s' ) ( s' ) (39) where s ' is as per Eq. (8). (Note that s ' is proportional to T0 ; the T0 terms cancel in the numerator and denominator and Teff is independent of T0.) The effective thermal contrast goes to zero as ( L a x 0 ) 0. 17

18 .7. Detection Decision Formulation For some standoff detection applications, making the correct plume absent / plume present detection decision can be more important than accurate estimation of the chemical vapor column density. The cost function in Eq. (18) facilitates detection decisions on the basis of a statistical F-test. The F value associated with the measured spectrum, ~ x, is where ( ~ x,θ ˆ ) ( ~ ˆ ) ( ~ C x ) ( ), θ ( ) 0 x = k 1 1 ~ ˆ C x, θ F (40) C is the cost function evaluated using the model parameters estimated with the Gauss-Newton algorithm and allowing all model parameters to vary and ( ~ x, θ ˆ ) evaluated with α fixed at zero. In contrast to C ( ~ x,θ ˆ ), calculation of ( ~ x, θ ˆ ) iterative computation. Under the background model defined in Section.3 C is cost function 0 0 C does not require an C ~ ~ ~ + β T ( ) [ ˆ 1 ] [ ˆ ] ˆ T x;θ = x Bβ D x Bβ β ˆ 0 0 (41) and the ˆβ 0 vector is calculated using Eq. (A.9) of Appendix A. Plume present is decided when the F value exceeds a user-specified threshold. For the linear signal model given by Eq. (30), the analogue to the F test is an adaptive subspace detector, the Adaptive Cosine Estimator (ACE) [4,5]. The ACE value associated with the measured spectrum x ~ is ( ~ x) T ( s Σˆ 1 ' [ ~ x µ ]) T ( s Σˆ 1 s )[ ~ T ' ' x µ ] Σˆ 1[ ~ x µ ] ( ) D = (4) ACE The ACE statistic can be regarded as cosine-squared of the angle between the test spectrum and the reference spectrum in noise-whitened, mean-subtracted signal space. The F value calculated in Eq. (40) is analogous to cotangent-squared of the spectral angle. The ACE value calculated in 18

19 Eq. (4) may be converted into an equivalent F value for direct comparison with the value calculated in Eq. (40), F = ( k ) D /( 1 D ) ACE 1. ACE ACE 3. Test Data For algorithm performance evaluation we simulated observations of fugitive emissions by augmenting plume-free measured spectra with synthetic plume signatures. The plume-free spectra were collected using an Adaptive Infrared Imaging Spectroradiometer-Wide Area Detector (AIRIS-WAD) [6,7]. The AIRIS-WAD sensor is an imaging Fabry-Perot spectrometer comprised of a pixel LWIR focalplane array (FPA) which views the far field through a rapidly tunable LWIR etalon. The sensor s optical system is configured to provide a 3 deg x 3 deg field-of-regard (. mrad per pixel IFOV). Spectra are recorded bandsequentially and consist of measurements at twenty (0) user-specified wavelengths in the 8 to 11 µm spectral region. The instrument s spectral resolution is ~0.08 µm, nominally 8 cm -1 FWHM at ν=1000 cm -1, and the lineshape is well-described by a Lorentzian function. The sensor is equipped with an internal blackbody source to facilitate real-time radiometric calibration of the sensor data. Figure shows a broadband IR representation of a datacube collected by the sensor. The broadband IR representation was generated by summing all twenty narrowband images in the datacube. Lighter pixels indicate higher radiance values. The scene is composed of low brush and compacted sand in the bottom half of the image and sky in the upper half. Figure 3 shows the average spectral radiance from the boxes marked sky, horizon, and ground in Figure. The spectrum of the ground region is very similar to the spectrum of a 30 K blackbody. The 19

20 sky spectrum is consistent with a low slant angle view to space and exhibits the characteristic ozone emission feature near 9.5 µm. We augmented AIRIS-WAD data with synthetic 1,1,1,-tetrafluoroethane (R-134a) spectra. R-134a is a freon widely-used in refrigeration systems and as a propellant for domestic and industrial applications. Data was augmented using the equation [ 1 τ ] o L + τ xˆ 0 + eˆ x p = p a p o (43) where τ p is the instrument-lineshape-averaged plume transmission, ˆx 0 is estimated noise-free background spectrum of the pixel as calculated using Eq. (A-10) of Appendix A, and ê is defined as the difference between the measured background spectrum and the estimated noisefree background spectrum, eˆ ~ x xˆ 0. The elements of τ p are: τ ( λ ) [ ρ κ( λ) ] g( λ λ ) = exp dλ (44) p s, The plume transmission at high resolution was calculated using an R-134a reference spectrum from the Pacific Northwest National Laboratories IR Spectral Database [8]. The instrument lineshape function used to evaluate Eq. (44) was Lorentzian with 0.08 µm FWHM, consistent with the experimentally-measured resolution function of the AIRIS-WAD sensor.. A useful characteristic of Eq. (43) is that it preserves the noise in the original data as plume becomes optically thick and thereby provides more realistic spectra for testing estimation algorithms than fully synthetic data with added Gaussian noise. As the plume transmission goes s zero x p L a + eˆ, i.e., as the plume becomes opaque the pixel spectrum becomes a noisy blackbody spectrum rather than a noise-free blackbody spectrum. Conversely, as the plume column density goes to zero its effective transmission goes to unity and x p = x 0, i.e., the output spectrum is equal to the original data if no plume is present. 0

21 The AIRIS-WAD datacube was segmented into four pixel horizontal quadrants for processing. The motivation for segmenting the data was twofold: 1) The AIRIS-WAD FPA has four separate readouts, each with somewhat different noise characteristics and ) division into four quadrants has the effect of partitioning the data into spectrally-similar subsets. The latter effect improves the fidelity of the infrared background model. The covariance matrix of each quadrant was calculated using a statistically-robust, Huber-type M-estimator [9]. The method used to determine the background subspace dimensionality is described in Appendix A. For the datacube depicted in Figure, the dimensionality of the quadrants ranged from m=4 to m=6. We exercised Eq. (43) to create synthetic datacubes with column density ranging from 0 to 591 ppmv-m (0 to.48 g/m ). For reference, 197 ppmv-m corresponds to OD=1.0 (base e) at 8.4 µm, the wavelength of strongest absorption in the R-134a spectrum, and the synthetic plumes varied from OD=0.0 to OD=3.0. Figure 4 shows the calculated transmission for 0 ppmv-m, 197 ppmv-m, and 591 ppmv-m R-134a plumes, i.e., plumes with peak optical densities of 0.10, 1.0, and 3.0. The plume temperature was set equal to the local air temperature, 98.0 K, for this simulation. Synthetic plumes were added to 64 pixel (horizontal) 5 pixel (vertical) regions in the scene and the column density was the same at each pixel where the plume signature was added. 4. Results and Discussion 4.1. General In this Section we compare the results obtained by applying the Gauss-Newton solver for detection and column density estimation with those obtained and the linear model given by Eq. (30). Prior to applying the algorithms to the plume-augmented AIRIS-WAD data we 1

22 verified that both the Gauss-Newton and clutter-matched filter/ace algorithms were properly implemented by processing purely synthetic data with additive Gaussian noise. The plumeaugmented AIRIS-WAD data was processed on a quadrant-by-quadrant basis and the Huber-type M-estimator was used to calculate the background covariance matrix of each quadrant. The motivation for using the M-estimator is that using a standard covariance calculation, i.e., giving equal weight to all spectra in a sample, results in a erroneous estimate of the background covariance when a plume is present in the sample. The M-estimator de-weights the contribution of statistically-anomalous spectra. Pixels where the plume signature is statistically-significant generally constitute statistical anomalies so the M-estimator generally provides a more accurate estimate of the true background covariance matrix. It is instructive to evaluate the scene in Figure and identify regions where the conditions are favorable for detection and, conversely, where they are not favorable for detection. Figure 5 shows the effective thermal contrast as a function of elevation. The effective thermal contrast was calculated for each pixel in the scene using Eq. (39) and Figure 5 shows the median for each row. The effective thermal contrast is ~0 K near the horizon (Row ~ 18) and one therefore expects detection statistics to be unfavorable in that region. The effective thermal contrast increases with elevation angle above the horizon and one expects detection statistics to be favorable above Row ~150 where Teff > 5 K. (The downward deviations in the vicinity of Rows 170 and 00 are due to clouds at those elevations.) These qualitative characterizations are addressed quantitatively in Figure 6, which shows uncertainty in the estimated column density as a function of elevation angle. The uncertainty was calculated for each pixel using Eq. (38) and Figure 6 shows the median for each row. We note that the modest discontinuities at Rows 64 and 19 are associated with boundaries between data quadrants. The discontinuity at Row 18 is

23 due primary to the transition from ground to low-angle sky background. The fact that it is also a quadrant boundary is a coincidence and is a minor contribution to the observed discontinuity. In the interest of comparing algorithm performance in favorable and unfavorable detection regions, we present results obtained by processing data with synthetic plumes added to the regions shown in Figure 7. The effective thermal contrast in Region 1 is.6 ± 0.5 K and the effective contrast in Region is 5.9 ± 0.6 K. (The variation is the 1σ standard deviation in T over the plume region, not the uncertainty in T.) 4.. Column Density Estimation In order to evaluate the accuracy and precision of the Gauss-Newton and linear model estimates of peak optical density, we calculate the median of the estimated values in the plume region, median{ αˆ i }, and the normalized median absolute deviation, a statistically-robust analogue of the sample standard deviation, ˆ σ ˆ α { ˆ α median{ ˆ α }} median i i = (45) median αˆ and ˆ σ ˆ α are very nearly equal to the sample mean and standard deviation when The { } i sample values are normally distributed; however, unlike the mean and standard deviation, { } i median αˆ and ˆ σ ˆ α are insensitive to low occurrence, highly anomalous values in the sample. median αˆ and ˆ σ ˆ α rather than the mean and standard deviation because we found We report { } i median ˆ ± ˆ generally yields a more accurate estimate of the range which incorporates that { α i} σ ˆ α ˆ ± 95% of the sample values than does mean { α } σ i. For the results we evaluated, we observed that the mean and median αˆ values were in excellent agreement and that ˆ σ ˆ α was typically 10-15% lower than the standard deviation. 3

24 Figures 8 and 9 show the median R-134a optical densities estimated using the Gauss-Newton solver and the clutter-matched filter. Figure 8 depicts the optical densities estimated in Region 1 and Figure 9 depicts the optical densities estimated in Region. The error bars in each figure correspond to σ ˆ α ± 1 ˆ. Optical densities may be converted to column densities by multiplying by ρ 0, 197 ppmv-m for R-134a. The solid symbols indicate the median OD estimated using the Gauss-Newton algorithm. The crossed open symbols indicate the median OD estimated using the clutter-matched filter. The black dashed line indicates perfect agreement between the actual and estimated OD values. As expected, the uncertainty in estimated OD is smaller in Region where the thermal contrast is greater. The variation in estimated αˆ values calculated using Eq. (45) is in reasonable agreement with the predictions of Eqs. (33) and (38). Eqs. (33) and (38) generally overestimated ˆ σ ˆ α by 30-70%. Results of simulations run using purely synthetic data with additive Gaussian noise suggest that the discrepancy is due to the fact that the noise in the test data is not precisely normally-distributed. The Gauss-Newton algorithm provides a more accurate estimate of column density than the linear model in all cases. As expected, the accuracy of the clutter-matched filter estimate degrades with increasing optical density. The systematic deviation of the Gauss-Newtonestimated optical densities from the Ideal line is due entirely to the approximation in Eq. (10). When the data is fit using the effective absorption spectrum for the appropriate ρ, κ e [ ] ρ ( ρ ) = ln τ ( ρ) /, rather than κ then all { } p median αˆ values all fall on the Ideal line. Figure 10 shows κ ( λ), the effective reference spectrum for OD=0.0, as well as ( λ) i κ for OD=1.0 (197 ppmv-m) and OD=3.0 (591 ppmv-m). The effective peak cross-section decreases by 9% from OD=0.0 to OD=1.0 and decreases by 9% from OD=0.0 to OD=3.0. The fact that the Gauss-Newton algorithm systematically underestimates αˆ values as the OD increases is e 4

25 consistent with the observed reduction in maximum effective absorption cross-section. The observed systematic underestimation of the OD using the Gauss-Newton algorithm suggests the true OD may be recovered by applying an OD-dependent correction factor to the estimated value. In principle, a correction faction could also be applied to the clutter-matched filter OD estimates; however, the fact that those OD values appear to approach a maximum value suggests that applying a correction factor would be problematic Plume Detection Statistics The standard performance metric for detection applications is the receiver operator characteristic (ROC) curve. Traditionally, a ROC curve is a plot of the probability of detection, P D, versus the probability of false alarm, P FA, where the (P D, P FA ) points which constitute the curve are calculated by varying the detection threshold from lowest to highest value. We use the terms detection rate (DR) and false alarm rate (FAR) here rather than P D and P FA because our results are data-derived rather than based on theoretical calculations of P D and P FA. We construct ROC curves as follows: 1. The n p F-values (or ACE values) in the plume region are put in ascending order: F,..., 1, F F n p. i 1/ / n. The DR for each F-value (or ACE value) is calculated as ( ) p 1 p DR 1 F-value s index, i = 1,,, n p. The range of DR is ( ) ( ) 1 n n., where i is the 3. The FAR corresponding to each DR is calculated by determining the number of F-values (or ACE values) in the off-plume region which are greater than or equal to F i. The p number of values exceeding F i is n,. For n 1, the false alarm rate is calculated as FA i FA, i i ( nfa, i 1/ ) nb FAR = /, where n b is the number of pixels in the off-plume region. If 5

26 n FA, i = 0, then we consider the (DR, FAR) point not to exist, so ( ) 1 FAR 1 ( n ) 1 n. b b Subtracting ½ from n p and n b when calculating DR and FAR is a convention for probability plotting which facilitates comparison with model distribution functions. Figures show the ROC curves calculated for Regions 1 and augmented with OD=0.1, 0.3, 1.0, and.0 plumes. The solid symbols correspond to points resulting from application of the Gauss-Newton algorithm. The crossed open symbols correspond to points results for application of the ACE algorithm. As the plume is optically-thin at OD=0.1 and 0.3 the two algorithms generate nearly identical ROC curves. We note that the ROC curves in Figure 11, OD=0.1, are all indicative of unfavorable plume detection statistics. Setting the detection threshold to achieve an 80% detection rate in Region would result in ~0% false positive rate in rest of the scene while setting the detection threshold to achieve 80% detection rate in Region 1 and would result in ~50% false positive rate. The ROC curves in Figure 1, OD=0.3, show somewhat more favorable detection statistics, particularly in Region. Setting the detection threshold to achieve an 80% detection rate in Region would result in ~0.3% false positive rate in rest of the scene and setting the detection threshold to achieve 80% detection rate in Region 1 and would result in ~30% false positive rate. Some separation between the Gauss-Newton and ACE curves is observed for OD=0.3 but the differences are modest. For both the OD=0.1 and OD=0.3 plumes the curves for Region are significantly more favorable than those for Region 1 because the effect thermal contrast is >x larger in Region than in Region 1. The ROC curves in Figure 13, OD=1.0, show significantly better detection statistics for the Gauss-Newton algorithm than the ACE algorithm in both Region 1 and Region. The [ αs] 1 αs exp approximation for the plume transmission is significantly less accurate than at 6

27 OD=0.1 and 0.3, so it is expected that the nonlinear estimator will outperform ACE-based detection. As was true at lower OD values, the detection statistics are significantly more favorable in the region of higher thermal contrast. In Region, >95% detection rate is achieved with a false positive rate of using Gauss-Newton algorithm while the false positive rate is approximately four orders of magnitude higher for the same detection rate in Region 1. Similarly, for ACE-based detection, the false alarm rate for a detection rate of 80% is approximately 350x lower in Region than it is in Region 1. Comparing the ROC curves for Gauss-Newton detection in Regions 1 with ACE-based detection in Region, while the degradation introduced by the [ αs] 1 αs exp approximation is apparent, it is less significant than the effect of enhanced thermal contrast in going from Region 1 to Region. Although the linear model which underlies the ACE detector is not precisely matched to the data, the ROC curve obtained by applying ACE to Region is still more favorable than the ROC curve obtained by applying the Gauss-Newton algorithm to Region 1. The ROC curves in Figure 14, OD=.0, show an even greater improvement in detection statistics for the Gauss-Newton algorithm relative to the ACE algorithm as [ αs] 1 αs exp is a poor approximation for the plume transmission near the wavelengths of strongest absorption. Comparing the ROC curve for Gauss-Newton detection in Regions 1 with ACE-based detection in Region, the degradation introduced by the [ αs] 1 αs exp approximation is more significant than the effect of enhanced thermal contrast in going from Region 1 to Region. With the OD increased to.0, the detection statistics in Region 1 become relatively favorable and for a detection rate of 80% the Gauss-Newton algorithm reduces the false positive rate by a factor of ~15 relative to ACE-based detection. The reduction in false alarm rate is even more pronounced in Region. One can also examine the differences in detection rate for a fixed false 7

28 alarm rate. For FAR=1 10-4, use of the Gauss-Newton algorithm increases the detection rate from ~10% to nearly 80% in Region 1. For FAR=1 10-5, use of the Gauss-Newton algorithm increases the detection rate from ~70% to ~99% in Region. The degradation of the ACE ROC curves with increasing OD can be understood by examining the deviation between the test data and the model spectra calculated using the maximum likelihood model parameters. Figure 15 shows spectra of pixels from Region 1 along with model spectra calculated using the maximum likelihood values of the model parameters. The original AIRIS-WAD spectrum is denoted by crossed open squares and the dashed line shows the corresponding model spectrum. (The difference between the best fits to the plumefree spectrum calculated using the Gauss-Newton and linear models is not observable on the scale of the graph.) For comparison, the solid squares denote the pixel spectrum after augmentation with an OD=3.0 R-134a plume. The solid line shows the spectrum calculated using the maximum likelihood parameters estimated using the Gauss-Newton algorithm and the dashed line shows the spectrum calculated using the maximum likelihood parameters estimated assuming the linear model, Eq. (30). While the differences appear modest in comparison to the range of spectral radiance values, the deviations between the data and the model spectra are systematic and readily discernable. Figure 16 shows the root-mean-squared (rms) residuals between the data and the model spectra in Region 1. For reference, the open diamonds show the rms deviation between the data and the model for no plume signature added to the region. The solid squares indicate the rms residuals for the linear model applied to Region 1 data augmented with a OD=3.0 (591 ppmv-m) plume. The rms deviation is ~3x greater between 8.4 and 9.0 µm and ~6x greater near 10.3 µm than it is for the plume-free spectra. The wavelengths where the rms deviation increases the 8

29 most correspond to wavelengths of strongest absorption, indicating a shortcoming in the fit model. For comparison, the crossed open squares indicate the rms deviation for the Gauss- Newton algorithm applied to the same data. While the rms deviation is slightly larger between 8.4 and 9.0 µm and near 10.3 µm as result of the difference between κ used for fitting and κ e ( ρ) used to create the test data, the residuals are in good agreement with rms deviations calculated for the plume-free data. This is the expected result when the signal model is consistent with the data Algorithm Convergence As the Gauss-Newton algorithm is iterative, it is necessary to define a termination criterion. Our termination criterion is the fractional change in the cost function given by Eq. (19). The algorithm terminates when the fractional change falls below a user-specified value, ε: C i+ ε (46) C The results presented in the preceding section were obtained using ε=0.01. In the event that the cost function increases from the i-th to (i+1)-th iteration the parameter values are restored to those from the i-th iteration and the algorithm terminates. Figure 17 shows number of iterations to convergence for Region augmented with OD=1.0,.0, and 3.0 plumes as well as the number of iterations observed when no plume was added. For reference, the solid bars show the number of iterations required for convergence in the plume-free region of the scene. Iteration zero corresponds to the initial guess at the maximum likelihood model parameters. The algorithm terminated after one iteration for ~55% of the plume-free pixels, i.e., the parameters were updated but the cost function did not change significantly from the initial guess, and the algorithm terminated after the second iteration for ~45% of the plume-free pixels. When applied i 9

30 to the original Region data, the algorithm terminated after one iteration approximately half of the time and after the second iteration the other half of the time, in good agreement with the plume-free region of the scene. The number of iterations required for convergence increases as the plume OD increases. For spectra augmented with a OD=1.0 plumes, all pixels required at least two iterations to converge; ~80% of the pixels require two iterations and ~0% required three iterations. When the plume OD is increased to.0, the algorithm required three iterations to converge for almost all pixels. A similar result was obtained for OD=3.0. Reducing ε to increases the number of iterations (from 3 to 4 for the OD=3.0 plume) however there was no significant effect on the estimated column density values. The observed differences at algorithm termination were on the order of 0.1% of the estimated values. For comparison, the uncertainties in the estimated values were typically several orders of magnitude larger than the differences observed by making the convergence threshold more stringent. 5. Summary and Conclusions We have presented a nonlinear optimal estimation method for detecting and characterizing chemical vapor fugitive emissions in a non-scattering atmosphere using passively-sensed LWIR spectra. The method integrates a parameterized signal model based on the RTE with a parameterized representation of covariance of the infrared background to create a probabilitybased cost function. The maximum likelihood model parameters are defined as those which minimize the cost function and are estimated using a Gauss-Newton algorithm. The algorithm formulation presented here presumes that the plume and air are in thermal equilibrium and that the air temperature is known; however, the algorithm may be easily modified to handle scenarios where the air temperature is not known. 30

31 For algorithm performance evaluation we simulated observation of fugitive emissions by augmenting plume-free spectra measured by an AIRIS-WAD sensor with synthetic R-134a plume signatures. The peak optical density of the synthetic plumes varied from OD=0.0 to OD=3.0. Results obtained by processing the simulated data indicate that the nonlinear estimator provides significantly more accurate estimates of chemical vapor column density and significantly more favorable detection statistics than matched-filter-based estimation when the vapor plume is optically-thick at one or more of the sensor observation wavelengths. This is because the signal model used for nonlinear estimation is based on the full clear air RTE, not an approximation which follows from the presumption of an optically-thin plume as do the clutter-matched filter and adaptive subspace detector. Finite instrument resolution introduced systematic error in column densities estimated using the Gauss-Newton algorithm but the effect was only significant for optical densities >>1.0 and the error is much smaller than that associated with clutter-matched filter estimates. For example, the Gauss-Newton algorithm underestimated column density by 17% at OD=3.0 whereas the clutter-matched filter underestimated it by ~70%. We note that while the nonlinear estimator provides significantly better results for optically-thick plumes, it produces the same result as a clutter-matched filter/adaptive subspace detector as the plume optical density approaches zero. The uncertainty in the column density estimated using the Gauss-Newton algorithm may be calculated from the Fisher information matrix which follows from the cost function. We observe that the uncertainties derived from the Fisher information matrix are typically 30-70% larger than the standard deviation of column densities estimated by processing the simulated sensor data. The discrepancy appears to be the result of non-gaussian noise in the originallymeasured plume-free spectra. In future implementations of the algorithm we plan to enable 31

32 estimation of air temperature, plume temperature, and atmospheric transmission effects as well as implement robust estimators to mitigate the effect of non-gaussian noise on estimates of maximum likelihood model parameters. 6. Acknowledgement This work was supported in part through contract no. HDTRA1-07-C-0067 with the Defense Threat Reduction Agency. 7. References 1. C. D. Rodgers, Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation, Rev. Geophys. Space Phys., 14, (4), (1976).. J. R. Eyre, Inversion of cloudy satellite sounding radiances by nonlinear optimal estimation. I: Theory and simulation for TOVS, Q. J. R. Meteorol. Soc., 115, (1989). 3. W. L. Smith, H. M. Woolf, and H. E. Revercomb, Linear simultaneous solution for temperature and absorbing constituent profiles from radiance spectra, Appl. Opt., 30, (9), (1991). 4. X. L. Ma, T. J. Schmit, and W. L. Smith, A nonlinear physical retrieval algorithm its application to the GOES-8/9 Sounder, J. Appl. Meteorol., 38, (1999). 5. X.L. Ma, Z. Wan, C.C. Moeller, W.P. Menzel, L.E. Gumley, and Y. Zhang, Retrieval of geophysical parameters from moderate resolution imaging spectroradiometer thermal infrared data: evaluation of a two-step physical algorithm, Appl. Opt., 39, (0), (000). 3

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35 1. R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis, Fifth Ed., (Prentice Hall, 00), Ch. 9, pp M. E. Tipping and C. M. Bishop, Probabilistic Principal Components Analysis, J. R. Statist. Soc. B, 61, part 3, 611-6, S. Kay, Fundamentals of Statistical Signal Processing: Volume 1, Estimation Theory, (Prentice-Hall, 1993) p S. Kraut and L. L. Scharf, The CFAR adaptive subspace detector is a scale invariant GLRT, IEEE Trans. Sig. Proc., 47, (9), (1999). 5. S. Kraut, L. L. Scharf, and L. T. McWhorter, Adaptive Subspace Detectors, IEEE Trans. Sig. Proc., 49, (1), 1-16 (001). 6. W. J. Marinelli, C. M. Gittins, B. R. Cosofret, T. E. Ustun, and J. O. Jensen, Development of the AIRIS-WAD multispectral sensor for airborne standoff chemical agent and toxic industrial chemical detection, Proc. of the Meetings of the Mil. Sens. Symp. Specialty Groups on Passive Sensors; Camouflage, Concealment, and Deception; Detectors; and Materials, Charleston, SC, Feb Available through Defense Technical Information Center (DTIC), document ref. no. ADA W. J. Marinelli, C. M. Gittins, A. H. Gelb, and B. D.Green, Tunable Fabry-Perot etalonbased long-wavelength infrared imaging spectrometer, Appl. Opt., 38, (1), (1999). 8. S. W. Sharpe, T. J. Johnson, R. L. Sams, P. M. Chu, G. C. Roderick, and P. A. Johnson, Gas-phase databases for quantitative infrared spectrometry, Appl. Spectrosc., 58, (004); DOE/PNNL Infrared Spectral Library Release 7.4, May

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37 Appendix A. Infrared Background Model In this work infrared background spectra, i.e., spectra corresponding to sensor views where no chemical vapor is present, are accounted for using a linear mixing model: x = µ + Bβ (A.1) where µ is the mean background spectrum, B is the k m dimensional matrix whose columns are the basis vectors used to span the data space and β is an m 1 vector of weight coefficients; m < k. Measured spectra are presumed to be subject to additive Gaussian noise: ~ x = x + e (A.) where e N( 0, D) D. The tilde denotes a noisy measurement and σ i is ~ ; = diag{ σ σ σ } 1,,..., k the 1σ standard deviation due to sensor noise and scene clutter. Estimation of D is described below. The B matrix follows from a regularization approximation of the sample covariance matrix. We summarize the method here and refer the reader to Refs. [] and [30] for additional detail. Regularization is implemented in two steps. The first step is to calculate the Principal Components decomposition of the noise-whitened sample covariance matrix: D / 1/ ΣD = UΛ U 1 T (A.3) where U is the m m matrix of eigenvectors and Λ is the diagonal matrix of eigenvalues, Λ = { λ λ,..., }. In this we estimate the D matrix as = { Σ } diag 1, λ k 1 [ ] 1 D diag and use the robust estimation method described by Tyler [9] to calculate Σ. The second step follows from the presumption that only the first m Principal Components and eigenvalues correspond to signal and the higher order components correspond to noise. This leads to an estimated sample covariance matrix 37

38 Σˆ T 1/ [ U ( Λ ε I ) U + εi ] D 1/ = D m m m m m (A.4) where U m is the k m matrix whose columns are the leading eigenvectors of 1 / 1/ D ΣD, { λ λ,..., } Λ m = diag 1, λ m, m I is the m m identity matrix and k 1 ε = λ (A.5) i k m i= m+ 1 Following Eqs. (A.4) and (A.5) the B matrix used in Eq. (A.1) is ( Λ εi ) 1/ B = D U (A.6) 1/ m m m As noted in Section.3, the β coefficients for the sample are presumed to be uncorrelated, have zero mean and unit standard deviation, ( 0, ) β ~ N I m (A.7) as would be the case when applying PCA to multivariate normal distributed data. The maximum likelihood β vector for an observed spectrum ~ x minimizes the cost function: ~ 1 ~ T T ( x Bβ) ( D) ( x Bβ) β β C = ε + (A.8) where the first term on the righthand side corresponds to the deviation between the data and the model and the β T β term follows from the prior distribution given by Eq. (A.7). Following the definition of B in (A.6), the maximum likelihood β vector is 1/ ( Λ I ) U Dˆ 1/ ( x µ ) ˆ 1 T β = Λ m m ε m m (A.9) ~ The corresponding maximum likelihood noise-free background spectrum is 1/ 1 T 1/ [ D U ( I Λ ) U D ]( ~ x µ ) µ x ˆ = m m ε m m + (A.10) Note that as m k, ε Λ 1 0 and xˆ x, i.e., if all Principal Components are deemed m significant, then the maximum likelihood spectrum is equal to the input spectrum. Conversely, 38

39 for a data set dominated by noise ε Λ I and xˆ µ, i.e., when the data is dominated by 1 m m noise then the maximum likelihood spectrum is equal to the sample mean. Determination of m, the dimensionality of the background subspace, is an information theory problem. The key criterion for any parameterized background model to be effective for chemical plume detection and characterization is that the dimensionality of the signal subspace must be much less than the number of bands in the measured spectra. We attempted to determine the number of statistically-significant signals in the data using information theoretic criteria; specifically, the Akaike Information Criterion (AIC) and Minimum Description Length (MDL) criteria [31]. (We note that the significance criterion in Appendix A of Ref. [] is equivalent to the MDL significance criterion.) The AIC and MDL criteria are appealing in that each is a function of the calculated eigenvalues and are thereby simple to evaluate. Unfortunately, we applied the AIC and MDL criteria to multiple datacubes and observed that neither criterion provided either stable, plausible estimates of the number of statisticallysignificant PCs; m k was a typical result even though the deviation between the model and the data changed little after the first several basis vectors. For this work we determined which basis vectors were statistically-significant by applying an F-test. For a model spectrum calculated using (m-1) basis vector, the F-test for statistical significance of the m-th basis vector is: F ( ) ( ) ( ) T ˆ 1 x xˆ ( ˆ ) ( ) ( ) m 1 D x x m 1 x; m = k m 1 1 T ˆ ˆ 1 x x ˆ m D x x m (A.11) ˆ where x m 1 is the maximum likelihood spectrum calculated using (m-1)-principal components and xˆ m is the maximum likelihood spectrum calculated using m-principal components. For normally-distributed noise, the calculated F-values follow an F-distribution with k-m-1 degrees 39

40 F x; m ~ F. In this work we decide statistical significance based on the of freedom, ( ) 1, k m 1 cumulative distribution of F-values. When <5% of the calculated F-values exceeded the F-value corresponding to 95% significance we considered the principal component statistically significant. We make no claim that this test is optimal for determining the model order but note that: 1) it produced the correct results using synthetic test data sets containing Gaussian additive noise and ) it produces seemingly reasonable results using real datacubes where noise is not precisely Gaussian. It would be far more computationally-efficient to use an eigenvalue-based method such as the AIC or MDL for determining model order; however, we were not able to identify one which performed reliably with the data of interest. As noted in Section 3, datacubes were processed on a quadrant-by-quadrant basis. Figure A.1 shows the fraction of spectra passing the F-test as a function m for the quadrant in of the datacube in which the synthetic spectra were embedded (upper middle). For that quadrant m=6. Figure A. shows the rms residuals in each band as a function of m for m=4-7 for the original (plume-free) datacube. Note that basis vectors have decreasing effect with increasing m. 40

41 8. Figure Captions Figure 1. Stratified atmosphere model. Each layer defined to have uniform temperature (T i ), pressure, and chemical composition; layer transmission is τ i. Chemical vapor plume of interest is Layer p. Figure. Gray scale representation of AIRIS-WAD datacube. Lighter pixels indicate higher average radiance values; average radiance calculated over all twenty spectral bands acquired by the sensor. Representative sky, horizon, and ground regions are indicated by the white boxes and black box, respectively. Figure 3. Radiance spectra corresponding to the sky, horizon, and ground regions in Figure. Spectra shown are the average of all pixels in the identified region. Figure 4. Calculated transmission spectra of 0, 197, and 591 ppmv-m R-134a plumes. High resolution spectra have peak optical density of 0.1, 1.0, and 3.0 (base e), respectively. The thick lines indicate the spectra calculated using Beer s law and the R-134a spectrum from the PNNL database. The thin dotted lines indicate the effective transmission which results from convolving the high resolution spectrum with a 0.08 µm FWHM Lorentzian lineshape function. The lower resolution spectra were used to augment AIRIS-WAD data. Figure 5. Effective thermal contrast between the local air temperature and the effective radiometric temperature of the background. Calculated median depicted in Figure. Teff for each row in scene Figure 6. Uncertainty in the estimated column density as a function of elevation angle. Plot shows median for row in scene depicted in Figure ; uncertainty calculated using Eq. (36). 41

42 Figure 7. Locations where synthetic R-134a plumes were added to AIRIS-WAD data. The effective thermal contrast in Region 1 is.6 ± 0.5 K and the effective contrast in Region is 5.9 ± 0.6 K. Figure 8. R-134a optical densities estimated in Region 1 of Figure 7. Black circles indicate median OD estimated by the Gauss-Newton algorithm. Open circles indicate median OD estimated using the linear signal model given by Eq. (31). The error bars in correspond to ± 1σ variation in estimated column density calculated using Eq. (49). Figure 9. R-134a optical densities estimated in Region of Figure 7. Black circles indicate median OD estimated by the Gauss-Newton algorithm. Open circles indicate median OD estimated using the linear signal model given by Eq. (31). The error bars in correspond to ± 1σ variation in estimated column density calculated using Eq. (49). Figure 10. Effective R-134a absorption cross-sections for OD=0.0, OD=1.0, and OD=3.0. The OD=0.0 spectrum is used for estimation of plume OD with the Gauss-Newton algorithm and linear estimator. Figure 11. ROC curves for OD=0.1 R-134a plumes added to Regions 1 and : = Gauss- Newton solver applied to Region, = ACE applied to Region, = Gauss-Newton solver applied to Region 1, = ACE applied to Region 1. Figure 1. ROC curves for OD=0.3 R-134a plumes added to Regions 1 and : = Gauss- Newton solver applied to Region, = ACE applied to Region, = Gauss-Newton solver applied to Region 1, = ACE applied to Region 1. Figure 13. ROC curves for OD=1.0 R-134a plumes added to Regions 1 and : = Gauss- Newton solver applied to Region, = ACE applied to Region, = Gauss-Newton solver applied to Region 1, = ACE applied to Region 1. 4

43 Figure 14. ROC curves for OD=.0 R-134a plumes added to Regions 1 and : = Gauss- Newton solver applied to Region, = ACE applied to Region, = Gauss-Newton solver applied to Region 1, = ACE applied to Region 1. Figure 15. Representative spectra from Region 1 and best fits to data using linear and nonlinear models: = original spectrum, = original spectrum augmented with OD=3.0 R-134a plume. Figure 16. Rms fit residuals for Region 1: = fit to original data, = fit to data augmented with OD=3.0 R-134a plume using nonlinear estimator, = fit to data augmented with OD=3.0 R- 134a plume using linear model. Figure 17. Number of iterations required for Gauss-Newton algorithm to converge, convergence threshold = 0.01: =plume-free pixels, = Region with no plume added, = Region with OD=1.0 plume added, = Region with OD=.0 plume added, = Region with OD=3.0 plume added. Figure A.1. Fraction of spectra passing F-test, Eq. (A.11), as function of number of basis functions used to model data. Pass criterion is F value for 5% of the spectra exceed the F value for 95% significance. Data corresponds to upper middle quadrant of scene in Fig.. Figure A.. RMS residuals between model and data as function of m for m=4-7. Six basis vectors were deemed to be statistically-significant using the F-test criterion. 43

44 9. Figures Fig

45 Fig.. 45